Working Group in the History and Philosophy of
Logic, Mathematics, and Science

2003-4

October 08, 2003, 6-7:30 PM in 234 Moses Hall

Erich Reck (U.C. Riverside, Philosophy)

Fregean versus Neo-Fregean Conceptions of Numbers

In recent neo-logicism, Frege’s conception of the natural numbers has been replaced by a new, modified variant. In this talk, I explore how Fregean this new conception is by discussing some of the original motivations and gradual modifications of Frege’s own approach. I also point towards an alternative way in which its core might be updated and saved from contradiction.

October 22, 2003, 6-7:30 PM in 234 Moses Hall

Aldo Antonelli (U.C. Irvine)

First-Order Quantifiers

In this talk we survey a number of results concerning first-order quantifiers and report on some work in progress. After briefly introducing generalized quantifiers, we identify some of their interesting properties and address the issue of their expressive power. Finally, we take up the question of the sense in which first-order quantifiers, similarly to second-order ones, might be open to a “non-standard” interpretation.

November 12, 2003, 6-7:30 PM in 234 Moses Hall

Alan Hájek (Cal Tech)

Waging War on Pascal’s Wager

Pascal’s Wager is simply too good to be true – or better, too good to be sound. There must be something wrong with Pascal’s argument that decision-theoretic reasoning shows that one must (resolve to) believe in God, if one is rational. No surprise, then, that critics of the argument are easily found, and they have attacked it on many fronts. For Pascal has given them no dearth of targets.

Virtually all of the Wager’s critics have directed their campaigns against its premises. Other authors have rallied to its defense, buttressing those premises. I will argue that they are fighting a lost cause: the Wager is simply invalid. This motivates a search for reformulations of the original argument that are valid, while upholding its spirit. I will offer four such reformulations, each of which finesses the decision matrix of the Wager, and in particular its problematic invocation of ‘infinite utility’. Yet these reformulations fall too, albeit for a different reason. This, in turn, might prompt advocates of the Wager to conduct another search for still further reformulations. However, I will argue that such a search is likely to be futile. When we examine what is at the root of the failure of the original Wager, and of the reformulations that I offer, we realize that their failures are symptomatic of a deep problem that any variant of the Wager must overcome. I will present a dilemma for all such variants, and conclude that their prospects for success are dim.

November 24, 2003, 6-7:30 PM in 234 Moses Hall

Andrew Arana (Stanford University, Philosophy)

Purity in Early Modern Geometry

Pappus’ division of problems into plane, solid, and linear, was widely influential in early modern geometry. Also influential was the related teaching that, e.g., plane problems should be solved by constructing plane curves, not solid or linear curves. As is well known, Descartes revised Pappus’ ordering of problems, by distinguishing instead between the geometrical and the mechanical. Nevertheless, Descartes maintained Pappus’ teaching that problems ought to be solved by means no more complex than the problem being solved. I have identified these teachings as instances of purity constraints. Roughly speaking, a solution is pure if it uses methods ‘close’ or ‘akin’ to the problem being solved. Purity constraints were well known in antiquity and the Middle Ages through Aristotle’s teachings on metabasis, but Descartes’ purity should be distinguished from Aristotle’s. I will explain why this is. My aim is to characterize the epistemic advantages of Descartes’ purity constraint, and evaluate those advantages over the advantages of impurity.

December 10, 2003, 6-7:30 PM in 234 Moses Hall

Robert May (UC Irvine)

The Essential Proposition: Frege on Identity Statements

Frege’s discussion of identity statements, especially in “On Sense and Reference,” is among his most famous, and the puzzle that swirls around them, of the informativeness of “Hesperus is Phosphorus” versus the triviality of “Hesperus is Hesperus,” bears, in the contemporary literature, the moniker “Frege’s Puzzle.” Discussions of Frege’s treatment of identity statements, however, often circulate around a well-tended myth that goes something like this. In Begriffsschrift, a youthful Frege presented an initial metalinguistic theory of identity statements; by the time of “On Sense and Reference,” Frege realized it was fatally flawed and replaced it with his storied mature theory in terms of sense and reference. Frege’s interest in identity statements, on this story, was part of the justification of a philosophical theory of meaning; answering why “Hesperus and Phosphorus” and “Hesperus is Hesperus” don’t mean the same thing led Frege to posit a substantive account of the meaning of expressions - the doctrine of sense and reference - and it is this doctrine that constitutes Frege’s enduring contribution. Understood this way, Frege’s discussion of identity statements gives rise to a common reading of “On Sense and Reference” on which it is a founding document in the canon of the philosophy of language.

Like all myths, this one contains a germ of truth; Frege did initially present a metalinguistic ac count, only to reject it in favor of an account in terms of sense and reference. And he does take the sense/reference distinction to be a substantive account of content. But nevertheless, the myth is seriously misleading about the underlying reasons for Frege’s interest in identity statements. That the myth invites us to read “On Sense and Reference” as occupying a position outside the mainstream of Frege’s oeuvre in indicative of the problem, for it detaches Frege’s discussion of identity statements from his central research project. But this, I will propose, is to get things just the wrong way around; it is identity statements themselves that are of primary interest to Frege. And the reason for this is centrally embedded in the logicist program, for such propositions are essential to establishing one of the central claims of the program, that numbers are, as Frege puts it, self-subsistent objects. The puzzle of identity statements called for solution just in order to maintain this result.

My goal in this paper is to explore Frege’s views of identity statements, and to place them in their proper philosophical and mathematical context. The main thesis to be explored, roughly put, is that the development of Frege’s views of identity statements from a metalinguistic to an objectual account tracks a change of emphasis from logical considerations in his earlier work to mathematical ones in the later work. In the course of these investigations, we will try to shed new and deeper light on a variety of questions - What role do identity statements play in the logicist program? Why did Frege initially adopt a metalinguistic view? What caused him to change his view to one in which identity statements express objectual identity? What is the significance of the puzzle, and what is it origin? How does “On Sense and Reference” fit into Frege’s oeuvre?

February 18, 2004, 6-7:30 PM in 234 Moses Hall

Karine Chemla (REHSEIS CNRS & Université Paris 7)

Technical terms and expressions related to proving the correctness of algorithms in the mathematics of ancient China

The earliest extant commentaries on the Han Canon The nine chapters on mathematical procedures (1st century BCE or CE), the one completed by Liu Hui in 263 and the one presented to the throne by Li Chunfeng in 656, systematically seek to establish the correctness of the algorithms described in the Canon. In previous publications, I have attempted to describe the practice of proof to which they bear witness. The intention of this paper is to show that the commentaries use technical terms and expressions in relation to their practice of proof. They highlight a reflection on operations that developed in ancient China and that may be of interest today for thinking about algorithms.

February 25, 2004, 6-7:30 PM in 234 Moses Hall

Paul Skokowski (Stanford University)

Structural Content: A Naturalistic Approach to Implicit Belief

This talk will examine how a system that learns can carry content that helps explain its behavior without an explicit representation. I begin by assuming a naturalistic theory of representational content, and push the theory to see how much it can explain. It soon becomes clear that a standard indicator approach won’t do for the implicit beliefs that operate in many, if not all, situations to which we apply belief/desire explanations. The model therefore needs to be extended to accommodate a new type of content for these situations.

March 10, 2004, 6-7:30 PM in 234 Moses Hall

Michael Detlefsen (University of Notre Dame)

Traces of Formalism

Historically important variants of formalism are identified and historically significant criticisms of them considered. A restricted version of formalism (one that does not see the whole of mathematics as formalist in character) is formulated and defended.

April 07, 2004, 6-7:30 PM in 234 Moses Hall

Ric Otte (UC Santa Cruz)

Managing Beliefs with Bayesian Conditionalization

Bayesianism is often construed as a theory about how we should manage our beliefs, while avoiding issues that are central to traditional epistemological theories. Central to Bayesianism is the rule of conditionalization, which is presented as a requirement of rational belief change. I will present an example which is problematic for the view that conditionalization is a necessary condition of rational belief change. This example also raises problems for the view that we should use conditionalization to manage our beliefs in pursuit of some ideal. As a result, I will argue that we generally cannot use conditionalization to manage our beliefs. I will then argue that this problem rises because of a confusion of external and internal factors within Bayesianism.

April 28, 2004, 6-7:30 PM in Howison Library

Miachel Friedman (Stanford University)

Physics, Philosophy, and the Foundations of Geometry: Einstein and the Logical Empiricists

The logical empiricists took late nineteenth-century work in the foundations of geometry by Riemann, Helmholtz, Poincare, and Hilbert as an inspiration for their distinctive philosophy of geometry. They also appealed centrally to Einstein’s general theory of relativity, which they viewed both as a culmination of the nineteenth-century foundational work in question and as fundamentally in agreement with their philosophical approach. Einstein’s famous paper “Geometry and Experience” was paradigmatic here. I examine the main argument of this paper against the late nineteenth-century mathematical background and show that it was fatefully – but also understandably – misunderstood by the logical empiricists. At the same time, however, this examination helps us appreciate the extraordinarily rich and intricate interaction between physics, philosophy, and the foundations of geometry which actually led to Einstein’s theory.